Principal component decomposition of non-parametric tests

Abstract

Let ϕ denote an arbitrary non-parametric unbiased test for a Gaussian shift given by an infinite dimensional parameter space. Then it is shown that the curvature of its power function has a principal component decomposition based on a Hilbert-Schmidt operator. Thus every test has reasonable curvature only for a finite number of orthogonal directions of alternatives. As application the two-sided Kolmogorov-Smirnov goodnessof-fit test is treated. We obtain lower bounds for their local asymptotic relative efficiency. They converge to one as α↓0 for the directionh0(u)=sign(2u−1) of the gradient of the median test. These results are analogous to earlier results of Hajek and Sidak for one-sided Kolmogorov-Smirnov tests.